Let f(x) = (2x^4 - 8x^3 + x^2 + 7x - 3)/(x^2 + x - 2). Give a polynomial g(x) so that f(x) + g(x) has a horizontal asymptote of 0 as x approaches positive infinity.

Guest Nov 15, 2020

#1**+2 **

This seems trickier than it really is.......

A rational function will have a horizontal asymptote of 0 whenever the degree of the polynomial in the numerator is less than the degree of the ploynomial in the denominator

So....all we need to have g(x) be is something where we change the signs on the 2x^4, -8x^3 and x^2 terms in the original polynomial (keeping the same denominator)

So....let g(x) be

(-2x^4 + 8x^3 - x^2 )

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(x^2 + x - 2)

When we add this to f(x) we get

(7x - 3)

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(x^2 + x - 2)

And this function will have a horizontal asymptote of 0 as x approaches infinity

CPhill Nov 15, 2020